We compute the
partition function on the hemisphere of a class of two-dimensional (2,2)
supersymmetric field theories including gauged linear sigma models. The
result provides a general exact formula for the central charge of the
D-brane placed at the boundary. It takes the form of Mellin-Barnes
integral and the question of its convergence leads to the grade
restriction rule concerning branes near the phase boundaries. We find
expressions in various phases including the large volume formula in

We apply supersymmetric localization to N=(2,2) gauged linear sigma
models on a hemisphere, with boundary conditions, i.e., D-branes,
preserving B-type supersymmetries. We explain how to compute the
hemisphere partition function for each object in the derived category of
equivariant coherent sheaves, and argue that it depends only on its K
theory class. The hemisphere partition function computes exactly the
central charge of the D-brane, completing the well-known formula

I will discuss a class of limiting points in the moduli space of d=2
(2,2) superconformal field theories. These SCFTs arise as IR limits of
"hybrid" UV theories constructed as a fibration of a Landau-Ginzburg
theory over a base Kaehler geometry. A significant generalization of
Landau-Ginzburg and large radius geometric limit points, the hybrid
theories can be used to probe general features of (2,2) and (0,2) SCFT
moduli spaces.

Mathieu Moonshine is a striking and unexpected relationship between the
sporadic simple finite group M24 and a special Jacobi form, the elliptic
genus, which arises naturally in studies of nonlinear sigma models with
K3 target. In this talk, we first discuss its predecessor (Monstrous
Moonshine), then
discuss the current evidence in favor of Mathieu Moonshine. We also
discuss extensions of this story involving `second quantized mirror
symmetry,' relating heterotic strings on K3 to type II strings on
Calabi-Yau threefolds.